* Ch. 15 Supplementary Exercises, problems 1, 5, & 6 *
1. Let n ≥ 2. If xi is a Boolean variable for all 1 ≤ i ≤ n, prove that
a) (x1 + x2 + ・ ・ ・ + xn) = x1x2 ・ ・ ・ xn * b) (x1x2 ・ ・ ・ xn) = x1 + x2 + ・ ・ ・ + xn *
5. Let B be a Boolean algebra that is partially ordered by ≤. If x, y, z ∈ B, prove that x + y ≤ z if and only if x ≤ z and y ≤ z. * * 6. State and prove the dual of the result in Exercise 5. * * Ch. 15 Exercise 15.1, problems 1, 2, 4, 5, 9, 12, & 15, *
1. Find the value of each of the following Boolean expressions if the values of the Boolean variables w, x, y, and z are 1, 1, 0, and 0, respectively.
a) xy + x y b) w + xy c) wx + y + yz * d) (wx + yz) + wy + (w + y)(x + y) *
2. Let w, x, and y be Boolean variables where the value of x is 1. For each of the following Boolean expressions, determine, if possible, the value of the expression. If you cannot determine the value of the expression, then find the number of assignments of values for w and y that will result in the value 1 for the expression.
a) x + xy + w b) xy + w * c) xy + xw d) xy + w *
4. a) Find the fundamental conjunction made up from the variables w, x, y, z, or their complements, where the value of the conjunction is 1 precisely when
i) w = x = 0, y=_ z = 1. ii) w = 0, x = 1, y = 1, z = 0. iii) w = 0, x = y = z = 1. iv) w = x = y = z = 0.

b) Answer part (a) this time for fundamental disjunctions, instead of fundamental conjunctions, where the value of each fundamental disjunction is 0 precisely for the stated values of w, x, y, z.

5. Suppose that f : B3 →B is defined by f (x, y, z) = (x + y) + (xz).
a) Determine the d.n.f. and c.n.f. for f .
b) Write f as a sum of minterms and as a product of maxterms (utilizing binary labels). *
9. Let Bn →B. If the d.n.f. of f has m fundamental conjunctions and its